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  • Orthogonality

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  • Definition of angle between vectors
  • Definition of orthogonal vectors
  • Definition of parallel vectors
  • Two vectors are orthogonal if and only if the Pythagorean Theorem holds.
  • Definition of a vector being orthogonal to a subspace
  • Definition of orthogonal complement of a subspace
  • The orthogonal complement of a subspace is a subspace.
  • The direct sum of a subspace and its orthogonal complement is the whole space.
  • A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.
  • The null space of a matrix is the orthogonal complement of the column space.
  • Definition of orthogonal set of vectors
  • Definition of orthonormal set of vectors
  • An orthogonal set of nonzero vectors is linearly independent.
  • Definition of orthogonal basis of a (sub)space
  • Definition of orthonormal basis of a (sub)space
  • A matrix A with real entries has orthonormal columns if and only if A inverse equals A transpose.
  • A matrix with real entries and orthonormal columns preserves norms.
  • A matrix with real entries and orthonormal columns preserves dot products.
  • Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.
  • A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.
  • Description of the Gram-Schmidt process
  • The Gram-Schmidt process converts a linearly independent set into an orthogonal set.
  • Definition of Gram-Schmidt process
  • Definition of orthogonal subspaces
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  1. A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.

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  • Linear systems and echelon matrices#23770
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  • Matrix addition is commutative and associative.#23463
  • Matrix adjoint is an involution.#23509
  • Matrix conjugation is an involution.#23466
  • Matrix describing a rotation of the plane#23887
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  • Matrix inverse is an involution.#23522
  • Matrix inverses are unique: if A and B are square matrices#23533
  • Matrix multiplication can be viewed as the dot product of a row vector of column vectors with a column vector of row vectors#23501
  • Matrix multiplication is associative.#23496
  • Matrix multiplication is distributive over matrix addition.#23497
  • Matrix multiplication is not commutative in general.#23499
  • Matrix representation of a composition of linear transformations is given by a matrix product#23862
  • Matrix-scalar multiplication is commutative#23472
  • Matrix-scalar product is commutative#23473
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